Some Archmage questions

[Viladin]

Branch Field: Yes, you use your full caster level.

Scholar Exploits: Yes.

Esoteric: For example, At 5th level, you have access to 3rd level spells, you can pick any spell from any spell list that you can cast, it becomes part of your repertoire. If its not normally a spell you can cast (outside of White/Black magic), it becomes 1 higher spell level. In this example, Enfire II would be a 3rd level spell instead of a 2nd level spell.

Alright good that's how I thought it would be but I wanted to make sure.

Great I know where my Moogle's FCB will be going and some feats to take now.

I can't seem to find the Enfire II spell, but lets say at 5th level where I can cast 3rd level spells I could pick a 2nd level Geomancer spell lets say Elemental Speech, I immediately learn and can cast it at 5th level but it would be treated as a 3rd level Scholar/Archmage spell and cost 3 MP right and would have any DCs as a 3rd level spell right?

Yes. That is correct.